Value at Risk: The Three Methods and the Loss It Never Sees

Value at risk puts one figure on a bad day: a loss a named book, over a named period, is not expected to pass on a named share of days. Take the Rs 3,600 crore held for trading portfolio at Vindhya Commercial Bank Limited, an invented bank, over one day at 99 per cent. Historical simulation says Rs 15.6 crore, variance covariance Rs 13.2 crore, Monte Carlo Rs 16.2 crore. Same book, same evening.

Two things have to be held together before that sentence is worth anything. The first is that value at riskA loss figure a stated book is not expected to exceed over a stated horizon on a stated share of outcomes, and nothing at all about the outcomes where it is exceeded. is not one number produced by one procedure. Value at risk is a question, and there are at least three respectable ways of answering it. The same securities on the same evening produce Rs 13.2 crore, Rs 15.6 crore and Rs 16.2 crore depending on who was asked. The second is that all three answers describe the same thing: a point in the range of possible outcomes. Every one of the three methods is estimating where a line falls, and not one of them is describing what lies beyond that line.

What does value at risk actually claim, and what does it refuse to say?

The claim is narrower than it sounds, and stating it precisely is half the work. Over a stated period, on a stated share of outcomes, the loss on a stated book is not expected to exceed a stated figure. At Vindhya Commercial Bank Limited, invented, the sentence reads in full: over one day, on 99 out of 100 days, the loss on the Rs 3,600 crore held for trading book is not expected to exceed Rs 15.6 crore. Notice the other day in the hundred: the sentence hands it over without a word. On that day the loss is larger. How much larger the sentence never says, and no amount of care in the arithmetic makes it say.

The shape of it appears in ordinary domestic life. A household that looks at what it spent each month for the last two years might say that in 99 months out of 100 the month does not cost more than Rs 90,000/-. The household statement is real, useful and checkable. The same statement is completely silent about the month the roof goes, the hospital admits somebody, or both land in the same fortnight. The statement was never about those months. The line sits at a busy month and goes no further. A reader who takes Rs 15.6 crore as the worst the trading book can lose has read a statement about ordinary bad days as though it were a statement about catastrophe. The bank has not made that claim, and the measure cannot support it.

What are the three choices sitting inside every value at risk figure?

Three, and none of them is optional. The first is the book: which positions are being measured. The second is the horizonThe period the figure covers. One day at this invented bank, and one of the three choices without which the figure means nothing.: over what period the loss is being imagined. The third is the confidence levelThe share of outcomes the figure is stated against, being 99 per cent at this invented bank, and the second of the three choices a reader must ask for.: on what share of outcomes the statement is being made. A value at risk quoted without all three attached is not a number, it is a rumour. Rs 15.6 crore over ten days is a different figure from Rs 15.6 crore over one day, and Rs 15.6 crore at 95 per cent is a different figure again. Somebody who compares two banks' headline figures without checking all three choices has compared nothing at all.

Which book is the measure on, and which Rs 22,800 crore is left outside it?

The whole investment book at Vindhya Commercial Bank Limited, invented, is Rs 26,400 crore, and value at risk here is measured on Rs 3,600 crore of it. The Rs 3,600 crore inside the measure is the held for trading portfolio. The available for sale book of Rs 8,400 crore and the held to maturity book of Rs 14,400 crore are banking book positions. Both are measured by the banking book rate measures instead, and neither sits inside the Rs 15.6 crore figure in any form. Rs 22,800 crore of the Rs 26,400 crore, being 86.4 per cent of this bank's investment book, sits outside the measure that carries its market risk limit, and that is a decision about which risk is being controlled rather than an omission. A reader who assumes value at risk covers everything a bank holds in securities has quietly stretched the coverage of the number more than sevenfold.

Value at Risk Methods: how does historical simulation build its number?

By refusing to invent anything and reordering what already happened. Historical simulationA method that reorders what actually happened over a stated window and reads the figure off the ordered results, assuming the window is representative of tomorrow. takes a stated observation windowThe stretch of past days a historical method uses, being 500 days at this invented bank, and the outer boundary of everything that method can possibly produce. of past days, applies the price moves of each of those days to the positions the bank is actually holding this evening, and produces one hypothetical result per day. Vindhya Commercial Bank Limited, invented, uses 500 days, so it produces 500 results. Order them from worst to best and the figure is read off at the point matching the confidence level. The method uses the distribution the market handed it, so it never asks what a distribution looks like. That is the whole appeal, and the fourth step is where the entire assumption is hiding: reading off that point is only meaningful if those 500 days are the kind of days tomorrow will be.

Where exactly a bank reads that point, and how it treats the days on either side of it, is set out in its own policy rather than being a fact of arithmetic. The shape matters: a historical method can only ever hand back a rearrangement of days that have already happened. It has no mechanism for producing a day of a kind the window does not hold, and it gives no warning that such a day exists.

Value at Risk Methods: what does variance covariance assume before it starts?

A shape. Variance covarianceA method that assumes a shape for the distribution of outcomes and computes the figure from volatilities and the relationships between positions. The method is fast and only as good as the assumed shape. does not reorder anything. The method takes how much each position tends to move, takes how the positions tend to move in relation to one another, assumes a shape for the spread of outcomes, and computes the point directly from those inputs. Because it is arithmetic on a handful of statistics rather than a pass over hundreds of reworked days, it is quick, it scales to very large books, and it can be rerun many times in a session. The price of that speed is that the answer is exactly as good as the assumed shape and no better. If real trading days pile up further out than the assumed shape allows for, the method understates the figure and reports it with the same composure it reports everything else.

Value at Risk Methods: what does Monte Carlo generate, and what does it inherit?

Monte CarloA method that generates a large number of possible futures from an assumed model and reads the figure off the results, inheriting every assumption inside that model. builds futures rather than borrowing the past or assuming a shape. A model generates a large number of possible next days, the positions are revalued in each one, and the results are ordered and read at the confidence level exactly as in the historical method. Vindhya Commercial Bank Limited, invented, generates 10,000 paths. The strength is that it copes with positions whose value does not move in a straight line with the market. The other two handle those positions awkwardly. The weakness is that every assumption inside the generating model is inherited whole, so 10,000 paths from a doubtful model give 10,000 doubtful answers rather than one. A large number of runs buys precision about the model. The runs buy nothing whatever about the world.

Why do three methods give three different answers on one book on one day?

Because they are answering the same question with three different kinds of trust, and the trust differs rather than the care. The size of the disagreement is the single fact most readers get wrong about this measure, and a reader who has formed an expectation before seeing the numbers is far harder to fool afterwards.

Here are the three, all on the Rs 3,600 crore held for trading book of Vindhya Commercial Bank Limited, invented, all for one day, all at 99 per cent, all measured on the same evening. Nothing in the position changed between them. The spread from lowest to highest is Rs 3.0 crore, or 22.7 per cent of the lowest figure. The spread is a property of the methods rather than of the book.

Two smaller facts are worth carrying out of that table. The mean of the three is exactly Rs 15.0 crore and the median is Rs 15.6 crore. The median is the figure this bank actually reports. So the method it uses sits above the average of the three and below the highest. The middle of three is a comfortable place to sit, and nothing about the arithmetic put this bank there. A bank running variance covariance on this identical book would publish a market risk figure a fifth smaller and would not be doing anything improper.

What does that spread do to a limit reported against the number?

The spread moves the control without moving the risk, and no consequence of the disagreement is sharper. Limit L5 at Vindhya Commercial Bank Limited, invented, caps trading book value at risk at Rs 18.0 crore. The cap is the bank's own, set by its board risk committee, and no outside body requires it. Against that same cap, limit utilisationA measured figure expressed as a percentage of its cap. A committee looks at that percentage when deciding whether a position needs a conversation. reads 73.3 per cent, 86.7 per cent or 90.0 per cent. The three readings span 16.7 percentage points of limit utilisation on one book on one evening, with not one rupee of position changing hands.

Sit with what that does to a committee. A desk at 90.0 per cent of its cap is a desk somebody is watching, with a conversation about trimming positions already half formed. A desk at 73.3 per cent is a desk with room to add. The two desks are the same desk, holding the same securities, on the same evening. The headroom to the cap reads Rs 1.8 crore, Rs 2.4 crore or Rs 4.8 crore, so the measured figure would have to rise 11.1 per cent, 15.4 per cent or 36.4 per cent before the limit is breached, depending on nothing but which method produced the numerator.

Which method does this bank use, and why is that choice written down?

Vindhya Commercial Bank Limited, invented, uses historical simulation, and limit L5 is measured on it. The recorded choice of method makes Rs 15.6 crore readable by anybody other than the person who produced it. Without it the figure is one of three, and the reader has no way of knowing which. The choice of method is therefore part of the limit and not part of the measurement. A written policy is where it belongs, not a spreadsheet somebody can improve on a quiet afternoon.

Follow the consequence. If the method sits in policy, moving from historical simulation to variance covariance is a decision: somebody proposes it, a committee considers what it does to every limit measured on the output, and the change is dated and recorded. If the method sits in a working file, the same move is a housekeeping change that drops reported utilisation from 86.7 to 73.3 per cent overnight and looks, in the monthly pack, exactly like a desk that has become more careful. Nothing about the securities differs between those two worlds. Only the paper trail differs, and the paper trail is the entire control.

Where did this measure come from, and why did it spread so quickly?

The spread came from one published document that made a single institution-wide figure practical, and that document deserves the name. The RiskMetrics technical document of 1994 set out a method and, just as importantly, a data set to run it on. Before it, a board asking how much the trading operation could lose on a bad day got a stack of position reports; after it, the same board got one figure computed the same way every morning. That is a governance change dressed as a statistical one, and it is the reason the measure went from a technique to an expectation in a handful of years. The artefact travelled rather than any institution, so the document rather than a firm carries the credit.

What is the loss it never sees, and can a higher confidence level reach it?

The loss it never sees is every loss past the line, and no confidence level reaches it. Picture the possible losses for one day laid out along a scale, the small ones on the left and the large ones running away to the right. Value at risk marks one point on that scale and says: this far, and on 99 days in a hundred no further. Everything to the right of that mark is left completely undescribed, and the measure has no vocabulary for it at all. A book that would lose Rs 16 crore on its worst imaginable day and a book that would lose Rs 160 crore on its worst imaginable day can both report Rs 15.6 crore, because the measure is looking at where the line falls and not at what stands behind it.

Now the move everybody tries. Raise the confidence level from 99 per cent to 99.9 per cent and the cut-offThe point on the scale of outcomes the figure sits at, which is what all three methods are estimating and the only thing any of them describes. slides further to the right. The reported figure gets larger and the sentence gets stricter. The stricter sentence stays exactly as silent. There is still a share of days beyond the new line, and the measure still says nothing whatever about them. A smaller blank region is still a completely blank region, and turning the dial has bought a bigger number rather than more information. Describing what lies beyond the line needs a measure built to average that region rather than to locate it. Such a measure is a separate subject.

Why is a bad observation window worse than a wrong number?

Because a wrong number can be checked and a missing kind of day cannot. Take the historical method at Vindhya Commercial Bank Limited, invented, at its word: it reorders 500 days that actually happened. The method cannot produce a kind of day those 500 days do not contain, and nothing in its output signals that such a day is missing. If the window happens to hold a quiet stretch, the measure is calm, well behaved and internally consistent, and it will stay that way right up to the morning it is useless. The arithmetic has nothing to compare the past against, so no diagnostic inside it can say the history supplied to it was unrepresentative.

The same shape appears in ordinary life. A household that has not had a medical emergency in five years builds its sense of a bad month from five years of months without one. The estimate is not careless. The estimate is built properly, from real data, by somebody sensible. The estimate is simply blind to a category of event the record does not contain, and the blindness is invisible from the inside. Taleb, The Black Swan, 2007, is the standing account of exactly this: the confidence a measured history gives about events the history does not hold. An assumed shape and a generating model are both statements about what kinds of day are possible, so the other two methods are not immune either. The historical method simply wears its boundary where it can be seen.

What did this bank's own year say about the measure?

The year said the claim was not met. Over 250 observation days the realised loss went past the measure on seven of them. A measure stated at 99 per cent expects about 2.5 such days in 250, so seven is 2.8 per cent of days against an expected 1.0 per cent. Seven against 2.5 is not a small miss, and the miss is a fact about this measure on this book in this year rather than a verdict on the method in general. Those seven days are numbered X1 to X7 in the bank's own records, and how that count is tested, what the pattern inside it means and what the bank did about it are worked through separately.

One detail is worth carrying even without the test. Four of the seven fell in consecutive pairs, X2 with X3 and X5 with X6. A measure that treats each day as independent of the day before does not expect pairs, so the clustering says something the count alone does not. At this level, the seven exceptions do one job: they show that the region past the line is populated, that it was populated more often than the sentence allowed for, and that no amount of care inside the measurement would have told anybody in advance.

What would anybody need to rebuild these three figures?

A return series, a set of volatilities, and a statement of how the positions move in relation to one another. Three finished results from an invented bank are what those inputs produce, and no substitute for them. Without those inputs not one of the three figures can be recomputed, checked or argued with. The gap is a limit of the illustration rather than a limit of the measure, and the inputs are most of the work in every real version of it.

The same gap sets up the right instinct for the real thing. When somebody hands an analyst a value at risk figure, the interesting questions are almost never about the last step of the arithmetic. The questions worth asking are these: which window, how long, how often refreshed, which shape assumed, which model generated the paths, and when was any of it last checked against what actually happened. The number is the smallest part of the number.

Which Rs 15.6 crore is this, and which ones is it not?

In this guide Rs 15.6 crore always means one thing: the one day value at risk at 99 per cent on the Rs 3,600 crore held for trading book of Vindhya Commercial Bank Limited, invented, measured by historical simulation over 500 days. The same digits turn up attached to completely unrelated things, so naming the object every time is not pedantry in a bank. In this invented bank's own records, a 5.0 per cent adverse move on the gross sum of currency positions FX1 to FX5 also lands on Rs 15.6 crore, and that is a scenario loss on a currency book rather than a distributional measure on a trading book. The one year cumulative repricing gap is Rs 15,600 crore. The gap shares the digits and is a banking book balance a thousand times the size. Operational incident I13 carries a net loss of Rs 15.4 crore, close enough to be misread by anybody reading at speed.

Four objects, one set of digits, and only one of them is the subject of this guide. Reading a risk pack means reading the noun before the number.

Who actually reads Rs 15.6 crore, and what do they do with it?

Four people read the same line for four different purposes, and only one of them is reading it as a risk figure at all. Knowing which of the four a reader is makes the difference between using the number and being used by it.

Notice what none of the four does. Not one of them reads Rs 15.6 crore as the most the desk can lose. The treasury reader treats it as a budget of measured risk, the risk reader treats it as a control percentage, the committee treats it as one line against the year's evidence, and the auditor treats it as a claim needing support. The moment somebody in the room starts using it as a ceiling on loss, every one of those four readings quietly breaks. A household version: the fact that the last two years of months never cost more than Rs 90,000/- is a useful planning figure and a terrible insurance policy.

Where does the standard come from, and what does an Indian bank have to do?

Two separate questions, and they have two separate answers that must be asked in that order. The measure itself is jurisdiction free: a book, a horizon, a confidence level, and three ways of estimating where the line falls. Nothing in the arithmetic is Indian or European or anything else. Any requirement attached to the measure is not jurisdiction free, and every such requirement has an issuing body and a date rather than a general truth behind it. The 99 per cent, the one day and the 500 day window are three choices Vindhya Commercial Bank Limited made and recorded in its own policy, and no requirement anywhere put them there.